Determinants and their applications in mathematical physics

Cover......Page 1
Determinants and Their Applications in Mathematical Physics......Page 4
Preface......Page 6
Contents......Page 10
1.2 Determinants......Page 16
1.3 First Minors and Cofactors......Page 18
1.4 The Product of Two Determinants --- 1......Page 20
2.2 Row and Column Vectors......Page 22
2.3.1 Basic Properties......Page 23
2.3.2 Matrix-Type Products Related to Row and Column Operations......Page 25
2.3.4 Alien Cofactors; The Sum Formula......Page 27
2.3.5 Cramer’s Formula......Page 28
2.3.7 The Derivative of a Determinant......Page 30
3.1 Cyclic Dislocations and Generalizations......Page 31
3.2.1 Rejecter and Retainer Minors......Page 33
3.2.2 Second and Higher Cofactors......Page 34
3.2.3 The Expansion of Cofactors in Terms of Higher Cofactors......Page 35
3.2.4 Alien Second and Higher Cofactors; Sum Formulas......Page 37
3.2.5 Scaled Cofactors......Page 38
3.3.1 A Grassmann Proof......Page 40
3.3.2 A Classical Proof......Page 42
Illustrations......Page 44
3.3.3 Determinants Containing Blocks of Zero Elements......Page 45
3.3.4 The Laplace Sum Formula......Page 47
3.3.5 The Product of Two Determinants — 2......Page 48
3.4 Double-Sum Relations for Scaled Cofactors......Page 49
3.5.2 The Cauchy Identity......Page 51
3.5.3 An Identity Involving a Hybrid Determinant......Page 52
3.6.1 The Jacobi Identity — 1......Page 53
3.6.2 The Jacobi Identity — 2......Page 56
3.6.3 Variants......Page 58
3.7.1 Basic Formulas; The Cauchy Expansion......Page 61
3.7.2 A Determinant with Double Borders......Page 64
Exercises......Page 65
4.1.1 Introduction......Page 66
4.1.2 Vandermondians......Page 67
Exercises......Page 68
4.1.3 Cofactors of the Vandermondian......Page 69
4.1.4 A Hybrid Determinant......Page 70
4.1.5 The Cauchy Double Alternant......Page 72
Exercises......Page 73
4.1.6 A Determinant Related to a Vandermondian......Page 74
4.1.8 Simple Vandermondian Identities......Page 75
4.1.9 Further Vandermondian Identities......Page 78
4.2 Symmetric Determinants......Page 79
4.3.1 Introduction......Page 80
4.3.2 Preparatory Lemmas......Page 84
Illustrations......Page 88
Exercises......Page 92
4.4.2 Factors......Page 94
4.4.3 The Generalized Hyperbolic Functions......Page 96
Illustrations......Page 98
4.5.1 Definition and Factorization......Page 100
4.5.2 Symmetric Toeplitz Determinants......Page 102
4.6.1 Definition and Recurrence Relation......Page 105
4.6.2 A Reciprocal Power Series......Page 107
4.6.3 A Hessenberg–Appell Characteristic Polynomial......Page 109
Exercises......Page 111
4.7.1 Introduction......Page 112
4.7.2 The Derivatives of a Wronskian......Page 114
4.7.3 The Derivative of a Cofactor......Page 115
4.7.5 Adjunct Functions......Page 117
4.7.6 Two-Way Wronskians......Page 118
4.8.1 Definition and the φm Notation......Page 119
4.8.2 Hankelians Whose Elements are Differences......Page 121
4.8.4 The Sum Formula......Page 123
4.8.5 Turanians......Page 124
4.8.6 Partial Derivatives with Respect to φm......Page 126
4.8.7 Double-Sum Relations......Page 127
Exercises......Page 129
4.9.1 The Derivatives of Hankelians with Appell Elements......Page 130
Exercises......Page 131
4.9.2 The Derivatives of Turanians with Appell and Other Elements......Page 134
Exercise......Page 137
4.10.1 The Generalized Hilbert Determinant......Page 138
Exercises......Page 140
4.10.2 Three Formulas of the Rodrigues Type......Page 142
4.10.3 Bordered Yamazaki–Hori Determinants — 1......Page 144
4.10.4 A Particular Case of the Yamazaki–Hori Determinant......Page 150
Exercises......Page 151
4.11.1 v-Numbers......Page 152
4.11.2 Some Determinants with Determinantal Factors......Page 153
4.11.3 Some Determinants with Binomial and Factorial Elements......Page 157
Exercises......Page 159
4.11.4 A Nonlinear Di.erential Equation......Page 162
Exercises......Page 167
4.12.1 Orthogonal Polynomials......Page 168
4.12.2 The Generalized Geometric Series and Eulerian Polynomials......Page 172
4.12.3 A Further Generalization of the Geometric Series......Page 177
4.13.1 Two Matrix Identities and Their Corollaries......Page 180
4.13.2 The Factors of a Particular Symmetric Toeplitz Determinant......Page 183
4.14 Casoratians --- A Brief Note......Page 184
5.1.1 Appell Polynomial......Page 185
5.1.2 The Generalized Geometric Series and Eulerian Polynomials......Page 187
Exercises......Page 188
5.1.3 Orthogonal Polynomials......Page 189
Exercises......Page 191
5.2.1 Three Determinants......Page 193
5.2.2 Four Lemmas......Page 195
5.2.3 Proof of the Principal Theorem......Page 198
5.2.4 Three Further Theorems......Page 199
5.3.1 A General Identity......Page 202
5.3.2 Particular Identities......Page 204
Exercises......Page 206
5.4.1 Introduction......Page 207
5.4.2 First Cofactors......Page 208
5.4.3 First and Second Cofactors......Page 209
5.4.4 Third and Fourth Cofactors......Page 210
5.4.5 Three Further Identities......Page 213
5.5.1 Continuants and the Recurrence Relation......Page 216
5.5.2 Polynomials and Power Series......Page 218
5.5.3 Further Determinantal Formulas......Page 224
5.6.1 Introduction......Page 226
5.6.2 Determinants with Binomial Elements......Page 227
5.6.3 Determinants with Stirling Elements......Page 232
Illustrations......Page 233
5.7.1 Definition and Taylor Relations......Page 236
5.7.2 A Determinantal Identity......Page 237
5.8.1 Introduction......Page 241
5.8.2 Hankel Determinants with Hessenberg Elements......Page 242
5.8.3 Hankel Determinants with Hankel Elements......Page 244
5.8.4 Hankel Determinants with Symmetric Toeplitz Elements......Page 246
5.8.6 Bordered Yamazaki–Hori Determinants — 2......Page 247
5.8.7 Determinantal Identities Related to Matrix Identities......Page 248
6.1 Introduction......Page 250
6.2.1 The Dale Equation......Page 251
6.2.3 The Toda Equations......Page 252
6.2.5 The Korteweg–de Vries Equation......Page 254
6.2.6 The Kadomtsev–Petviashvili Equation......Page 255
6.2.8 The Einstein and Ernst Equations......Page 256
6.2.9 The Relativistic Toda Equation......Page 260
6.3 The Dale Equation......Page 261
6.4 The Kay-Moses Equation......Page 264
6.5.1 The First-Order Toda Equation......Page 267
6.5.2 The Second-Order Toda Equations......Page 269
6.5.3 The Milne-Thomson Equation......Page 271
6.6.1 A System With One Continuous and One Discrete Variable......Page 273
6.6.2 A System With Two Continuous and Two Discrete Variables......Page 276
6.7.1 Introduction......Page 278
6.7.2 The First Form of Solution......Page 279
6.7.3 The First Form of Solution, Second Proof......Page 283
6.7.4 The Wronskian Solution......Page 286
6.7.5 Direct Verification of the Wronskian Solution......Page 288
6.8.1 The Non-Wronskian Solution......Page 292
6.8.2 The Wronskian Solution......Page 295
6.9.1 Introduction......Page 296
6.9.2 Three Determinants......Page 297
6.9.3 Proof of the Main Theorem......Page 300
6.10.2 Preparatory Lemmas......Page 302
6.10.3 The Intermediate Solutions......Page 307
6.10.4 Preparatory Theorems......Page 310
6.10.5 Physically Significant Solutions......Page 314
6.11 The Relativistic Toda Equation --- A Brief Note......Page 317
The Kronecker Delta Function......Page 319
The Binomial Coeficient and Gamma Function......Page 320
Stirling Numbers......Page 321
Inversions, the Permutation Symbol......Page 322
Permutations Associated with Pfaffians......Page 324
A.3 Multiple-Sum Identities......Page 326
A.4 Appell Polynomials......Page 329
Appell Sets......Page 332
Multiparameter and Multivariable Appell Polynomials......Page 333
Exercises......Page 335
and Ln(x)......Page 336
Legendre Polynomials Pn(x)......Page 337
A.6 The Generalized Geometric Series and Eulerian Polynomials......Page 338
Examples......Page 341
A.8 Differences......Page 343
A.9 The Euler and Modified Euler Theorems on Homogeneous Functions......Page 345
Formulas Related to the Function
(x +
√
1 + x2)2n......Page 347
A.11 Solutions of a Pair of Coupled Equations......Page 350
Transformation γ......Page 352
Transformation ε......Page 353
Transformation ß (Ehlers)......Page 354
A Table of Contents......Page 356
Bibliography......Page 358
Index......Page 388